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PreprintCited by 21Quantum Computing Algorithms and ArchitectureQuantum many-body systemsQuantum Information and Cryptography

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Machine learning on quantum experimental data toward solving quantum many-body problems

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Kim, Dohun, Cho, Gyungmin·2023·Nature Communications
DOI 10.21203/rs.3.rs-3674601/v1PMC PMC11364692

Preprint — published as 10.1038/s41467-024-51932-3 · Paper page

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Abstract

Advancements in the implementation of quantum hardware have enabled the acquisition of data that are intractable for emulation with classical computers. The integration of classical machine learning (ML) algorithms with these data holds potential for unveiling obscure patterns. Although this hybrid approach extends the class of efficiently solvable problems compared to using only classical computers, this approach has been realized for solving restricted problems because of the prevalence of noise in current quantum computers. Here, we extend the applicability of the hybrid approach to problems of interest in many-body physics, such as predicting the properties of the ground state of a given Hamiltonian and classifying quantum phases. By performing experiments with various error-reducing procedures on superconducting quantum hardware with 127 qubits, we managed to acquire refined data from the quantum computer. This enabled us to demonstrate the successful implementation of classical ML algorithms for systems with up to 44 qubits. Our results verify the scalability and effectiveness of the classical ML algorithms for processing quantum experimental data.

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Fig. 1 Classical machine learning (ML) on quantum experimental data. a Conversion of the information represented in a quantum state into a classical form for the application of classical ML. The training data were the expectation values measured via the quantum computer or the classical shadows of the quantum state. b Predicting the properties of the ground state of a given Hamiltonian casts to regression in classical ML. The objective of the regression is to approximate the target function f(x) = Tr(Oρ(x)) as closely as possible by trained ML model using training data ({()xi,Tr(Oρ(xi))})i=1^Ndata. After training, when a new input xnew is given, the trained ML model outputs f^(xnew) as a guess for f(xnew). c In classification, the objective is to distinguish a particular quantum phase from others by training a ML model that gives rise to the phase boundary on a relevant space. After training, when a new state is given, the trained ML model assigns a phase of the state by considering its position relative to the phase boundary on the space.
Fig. 1 Classical machine learning (ML) on quantum experimental data. a Conversion of the information represented in a quantum state into a classical form for the application of classical ML. The training data were the expectation values measured via the quantum computer or the classical shadows of the quantum state. b Predicting the properties of the ground state of a given Hamiltonian casts to regression in classical ML. The objective of the regression is to approximate the target function f(x) = Tr(Oρ(x)) as closely as possible by trained ML model using training data ({()xi,Tr(Oρ(xi))})i=1^Ndata. After training, when a new input xnew is given, the trained ML model outputs f^(xnew) as a guess for f(xnew). c In classification, the objective is to distinguish a particular quantum phase from others by training a ML model that gives rise to the phase boundary on a relevant space. After training, when a new state is given, the trained ML model assigns a phase of the state by considering its position relative to the phase boundary on the space.CC BY-NC-ND · PubMed Central
Fig. 2 Predicting ground state properties. a Hamiltonian for random hopping system and Su–Schrieffer–Heeger (SSH) system. xi in Hhop(x) is sampled uniformly from [0, 2]. b Ground state preparation circuit. Gates labeled G and P represent a Givens rotation and a basis rotation for parity measurement, respectively. c Correlation matrix for random hopping system. Results from ML prediction using the Gaussian kernel and exact values from exact diagonalization (ED) are shown on the left and right, respectively. d Prediction error. Model performance is measured by the root-mean-square error (RMSE). The error bars represent the standard deviation of the RMSEs of each observable. The dotted red line means the average RMSE for the training data. Inset shows the relationship between the log(Ndata) and 1/RMSE. The solid lines indicate a linear fit to the data points. e Correlation matrix for 1D SSH system (v = 1, w = 1). Results from ML predictions using Gaussian kernel and exact values from ED are shown on the left and right, respectively. f Prediction error. RMSE* (=√(2/n(n−1)(∑)i≥jⁿ(⟨ai^(†)aj⟩−⟨ai^(†)aj⟩ML)²)) from different kernels while keeping v = 1 and varying w in 0.25 intervals from 0.25 to 1.75. g Edge correlation. The edge correlation 〈a1†an〉 is measured between sites located at either end of a chain. The dotted red line is the known phase boundary between the trivial and topological phases. The black curve indicates exact values obtained by ED. In d, f, g the green, blue, and orange points each indicate the results predicted by the ML model trained by a (modified) Dirichlet kernel, neural tangent kernel, and Gaussian kernel, respectively.
Fig. 2 Predicting ground state properties. a Hamiltonian for random hopping system and Su–Schrieffer–Heeger (SSH) system. xi in Hhop(x) is sampled uniformly from [0, 2]. b Ground state preparation circuit. Gates labeled G and P represent a Givens rotation and a basis rotation for parity measurement, respectively. c Correlation matrix for random hopping system. Results from ML prediction using the Gaussian kernel and exact values from exact diagonalization (ED) are shown on the left and right, respectively. d Prediction error. Model performance is measured by the root-mean-square error (RMSE). The error bars represent the standard deviation of the RMSEs of each observable. The dotted red line means the average RMSE for the training data. Inset shows the relationship between the log(Ndata) and 1/RMSE. The solid lines indicate a linear fit to the data points. e Correlation matrix for 1D SSH system (v = 1, w = 1). Results from ML predictions using Gaussian kernel and exact values from ED are shown on the left and right, respectively. f Prediction error. RMSE* (=√(2/n(n−1)(∑)i≥jⁿ(⟨ai^(†)aj⟩−⟨ai^(†)aj⟩ML)²)) from different kernels while keeping v = 1 and varying w in 0.25 intervals from 0.25 to 1.75. g Edge correlation. The edge correlation 〈a1†an〉 is measured between sites located at either end of a chain. The dotted red line is the known phase boundary between the trivial and topological phases. The black curve indicates exact values obtained by ED. In d, f, g the green, blue, and orange points each indicate the results predicted by the ML model trained by a (modified) Dirichlet kernel, neural tangent kernel, and Gaussian kernel, respectively.CC BY-NC-ND · PubMed Central
Fig. 3 Experimental results for distinguishing between symmetry protected topological (SPT) and trivial phases. a Trivial and SPT phases. When considering a set of gapped symmetric Hamiltonians only, the two phases cannot be connected by a constant depth symmetric local unitary (Path I). However, without consideration of the symmetry, there exists a constant depth local unitary circuit (Path II) connecting them. b Generation of data. The process involves applying two layers of symmetric local random unitary to a fixed-point state associated with a particular phase, followed by randomized measurement for classical shadows. Random gates used can be found in the Supplementary Note 5. c String order parameter (SOP) (Sab = ZaXa+1Xa+3…Xb-3Xb-1Zb) of various lengths measured by direct measurements on the quantum computer. The blue and orange dots correspond to the SPT and trivial phases, respectively. The error bars represent the standard deviation of 300 SOPs. d Classification errors. The chart shows the classification error of the ML model for cases with ℤ2⊗ℤ2, TRS, and without symmetry constraints. The error bars represent the standard deviation from 10 instances. e, f 2D representation of the test data. Each figure corresponds to ℤ2⊗ℤ2 (e) and time-reversal symmetry (TRS) (f). g Classification results. The background represents the expectation values of a SOP measured from the ground states of the HCI, and these values are used to determine the ground truth of the quantum phase. For the 40 randomly sampled test data (marked by circles), each phase predicted by the trained ML model is colored in blue for the SPT phase and orange for the trivial phase. The red dot represents a point where the model incorrectly predicted the phase.
Fig. 3 Experimental results for distinguishing between symmetry protected topological (SPT) and trivial phases. a Trivial and SPT phases. When considering a set of gapped symmetric Hamiltonians only, the two phases cannot be connected by a constant depth symmetric local unitary (Path I). However, without consideration of the symmetry, there exists a constant depth local unitary circuit (Path II) connecting them. b Generation of data. The process involves applying two layers of symmetric local random unitary to a fixed-point state associated with a particular phase, followed by randomized measurement for classical shadows. Random gates used can be found in the Supplementary Note 5. c String order parameter (SOP) (Sab = ZaXa+1Xa+3…Xb-3Xb-1Zb) of various lengths measured by direct measurements on the quantum computer. The blue and orange dots correspond to the SPT and trivial phases, respectively. The error bars represent the standard deviation of 300 SOPs. d Classification errors. The chart shows the classification error of the ML model for cases with ℤ2⊗ℤ2, TRS, and without symmetry constraints. The error bars represent the standard deviation from 10 instances. e, f 2D representation of the test data. Each figure corresponds to ℤ2⊗ℤ2 (e) and time-reversal symmetry (TRS) (f). g Classification results. The background represents the expectation values of a SOP measured from the ground states of the HCI, and these values are used to determine the ground truth of the quantum phase. For the 40 randomly sampled test data (marked by circles), each phase predicted by the trained ML model is colored in blue for the SPT phase and orange for the trivial phase. The red dot represents a point where the model incorrectly predicted the phase.CC BY-NC-ND · PubMed Central
Fig. 4 Experimental results for distinguishing between topologically ordered and trivial phases. a Schematic diagram for measurement-assisted state preparation method on the heavy-hexagonal lattice. The application of a sequence of unitary transformations and adaptive gates depending on the measurement results of some ancilla qubits was followed by randomized measurements for classical shadows. The blue shaded areas represent the X-plaquettes. b Generation of data in the trivial phase. States in this phase are generated by applying local random unitary to the product state at varying circuit depth (dLU). Detailed information about the local random unitary that was used is provided in the Supplementary Note 5. c 1D projection of data. Using kernel principal component analysis with the shadow kernel in Eq. (3), each data point was projected onto the first principal axis. d 1D projection of data without topological order.
Fig. 4 Experimental results for distinguishing between topologically ordered and trivial phases. a Schematic diagram for measurement-assisted state preparation method on the heavy-hexagonal lattice. The application of a sequence of unitary transformations and adaptive gates depending on the measurement results of some ancilla qubits was followed by randomized measurements for classical shadows. The blue shaded areas represent the X-plaquettes. b Generation of data in the trivial phase. States in this phase are generated by applying local random unitary to the product state at varying circuit depth (dLU). Detailed information about the local random unitary that was used is provided in the Supplementary Note 5. c 1D projection of data. Using kernel principal component analysis with the shadow kernel in Eq. (3), each data point was projected onto the first principal axis. d 1D projection of data without topological order.CC BY-NC-ND · PubMed Central
Fig. 5 Extracting a phase classifier from a trained ML model. a Illustrative diagram of ML procedure. In the first step, given a quantum state ρ, we used feature mapping φ(ρ) to reduce the dimensionality of the data. To achieve an error-resilient phase classifier fML(ρ), training data are generated by applying random unitary transformations to fixed-point states. b Classification error. The orange bars represent the results from the ML model trained on raw data without measurement error mitigation (MEM). The blue bars represent the classification error of the ML model trained by the data with MEM and the Renyi-2 Topological entanglement entropy (TEE). The error bar denotes the standard deviation of 100 instances of the test data set.
Fig. 5 Extracting a phase classifier from a trained ML model. a Illustrative diagram of ML procedure. In the first step, given a quantum state ρ, we used feature mapping φ(ρ) to reduce the dimensionality of the data. To achieve an error-resilient phase classifier fML(ρ), training data are generated by applying random unitary transformations to fixed-point states. b Classification error. The orange bars represent the results from the ML model trained on raw data without measurement error mitigation (MEM). The blue bars represent the classification error of the ML model trained by the data with MEM and the Renyi-2 Topological entanglement entropy (TEE). The error bar denotes the standard deviation of 100 instances of the test data set.CC BY-NC-ND · PubMed Central

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Metadata

Authors
Kim, D., & Cho, G.
Affiliations
Seoul National University (KR)
Corresponding author
  • Kim, Dohun
  • Cho, Gyungmin
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2023-12-12
Volume
15
Issue
1
Pages
7552-7552
ISSN
2041-1723 (ISSN-L)
DOI
10.21203/rs.3.rs-3674601/v1
PMC
PMC11364692
OpenAlex
W4402039629
Cited by
21
References
1
Citations by year
2026: 11 · 2025: 7 · 2024: 3
FWCI
3.09
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by-nc-nd)
  • Research Square (acceptedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by-nc-nd)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion, cc-by)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University) (cc-by)
Funders
  • Imperial College London
  • National Research Foundation (2019R1A5A1027055)
  • Seoul National University
  • Korea Basic Science Institute (2021R1A6C101B418)
  • National Research Foundation of Korea (2019M3E4A1080144, RS-2023-00283291, 2019R1A5A1027055, 2023R1A2C2005809, 2021R1A6C101B418, 2020M3H3A1110365, RS-2023-00207732, 2019M3E4A1080145, RS‐2024‐00413957)
  • Ministry of Science and ICT, South Korea (2019R1A5A1027055, 2021R1A6C101B418, 2020M3H3A1110365)
  • University of Illinois at Urbana-Champaign
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
Keywords
hybrid quantum-classical machine learning, quantum many-body problem, superconducting quantum processors, error mitigation, scalability

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Kim, D., & Cho, G. (2023). Machine learning on quantum experimental data toward solving quantum many-body problems. Nature Communications. https://doi.org/10.21203/rs.3.rs-3674601/v1